Another chemical alternative is based on success that Bermuda seed
growers have when they are discontinuing an old field and
converting it to lettuce, etc. production. They till the
field up in mid winter and leave the soil clumps and roots and
rhizomes exposed. Any winter freeze will help knock the grass
and kill more of it. Then usually they will till the soil a
couple of more times in winter to keep exposing new portions of
rhizomes and crowns.

If You Must Use Chemicals

You need to make sure the zoysia is growing juicy and lush
and green, before spraying. Very critical when trying to kill
any of the warm climate grasses, to make sure they are actively
growing or else the plant will not absorb much of the
Glyphosphate. Most people think they are trying to kill
the grass, so watering and feeding it is the last thing comes to
their minds. But, if the zoysia is stressed for water and
fertilizer or it is too early in the year so it is not growing
much, the spray is a waste of money and time as the grass plants
will not absorb the poison spray while they are semi- or fully
dormant.

And, even with all this, it is still most likely they
will need to spray two or more times to get full kill.
Spray once and leave the grass alone for a week or two to allow
full absorption and transfer to the roots of the poison. Then
rake and mow off the dead grass, and immediately water well and
deeply to see if any is left alive to re-grow. Repeat as
necessary. Once any remaining stems green up and are lush
again, spray again on those patches and repeat the wait
period. This is usually enough to wipe out any persistent warm
climate grass.

Some extension agents recommend (and I have heard this from turf
care professional also) to add some fertilizer to the spray
tank, and also some sort of a surfactant. The liquid
fertilizer (even dissolved Miracle Grow works) stimulates the
grass plants to absorb more poison, and to continue active
growth so the poison can act more effectively. The surfactant
(a few tablespoons of liquid dish soap is a cheap substitute)
will allow the spray to spread and stick on the leaf surface,
rather than just beading up and running off before it is
absorbed.

BOTTOM LINE: Is that Zoysia because of its deep and persistent
root system (Rhizomes) is VERY DIFFICULT to get a complete kill
on. Good Luck!